# StratonovichProcess

StratonovichProcess[{a,b},x,t]

represents a Stratonovich process , where .

StratonovichProcess[{a,b,c},x,t]

represents a Stratonovich process , where .

StratonovichProcess[,,{x,x0},{t,t0}]

represents a Stratonovich process with initial condition .

StratonovichProcess[,,,Σ]

uses a Wiener process , with covariance Σ.

StratonovichProcess[proc]

converts proc to a standard Stratonovich process whenever possible.

StratonovichProcess[sdeqns,expr,x,t,wdproc]

represents a Stratonovich process specified by a stochastic differential equation sdeqns, output expression expr, with state x and time t, driven by w following the process dproc.

# Details and Options   • StratonovichProcess is also known as Stratonovich diffusion or stochastic differential equation (SDE).
• StratonovichProcess is a continuous-time and continuous-state random process.
• If the drift a is an -dimensional vector and the diffusion b an × -dimensional matrix, the process is -dimensional and driven by an -dimensional WienerProcess.
• Common specifications for coefficients a and b include:
•  a scalar, b scalar a scalar, b vector a vector, b vector a vector, b matrix • A stochastic differential equation is sometimes written as an integral equation .
• The default initial time t0 is taken to be zero, and default initial state x0 is zero.
• The default covariance Σ is the identity matrix.
• A standard Stratonovich process has output , consisting of a subset of differential states .
• Processes proc that can be converted to standard StratonovichProcess form include OrnsteinUhlenbeckProcess, GeometricBrownianMotionProcess, ItoProcess, and StratonovichProcess.
• The stochastic differential equations in sdeqns can be of the form , where is \[DifferentialD], which can be input using dd . The differentials and are taken to be Stratonovich differentials.
• The output expression expr can be any expression involving x[t] and t.
• The driving process dproc can be any process that can be converted to a standard Stratonovich process.
• Properties related to StratonovichProcess include:
•  "Drift" drift term "Diffusion" diffusion matrix "Output" output state "TimeVariable" time variable "TimeOrigin" origin of time variable "StateVariables" state variables "InitialState" initial state values "KolmogorovForwardEquation" Kolmogorov forward equation (Fokker-Planck equation) "KolmogorovBackwardEquation" Kolmogorov backward equation "Derivative" Stratonovich derivative
• Method settings in RandomFunction specific to StratonovichProcess include:
•  "EulerMaruyama" Euler–Maruyama (order 1/2, default) "KloedenPlatenSchurz" Kloeden–Platen–Schurz (order 3/2) "Milstein" Milstein (order 1) "StochasticRungeKutta" 3‐stage Rossler SRK scheme (order 1) "StochasticRungeKuttaScalarNoise" 3‐stage Rossler SRK scheme for scalar noise (order 3/2)
• StratonovichProcess can be used with such functions as RandomFunction, CovarianceFunction, PDF, and Expectation.

# Examples

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## Basic Examples(1)

Define a process by its stochastic differential equation:

Simulate the process:

Compute the mean function:

Compute the covariance function:

## Scope(15)

### Basic Uses(9)

Define a Wiener process with drift and diffusion from the sde :

Directly convert from the parametric process:

Define a process , where :

Use differential notation to define the same process:

Define a vector process with output :

Using differential notation:

Define a vector process , where :

Using differential notation:

Define a vector process where :

Using differential notation:

Define a process driven by two correlated Wiener processes:

Define a scalar process corresponding to the sde :

Define vector process and corresponding to the sde and :

Define a process corresponding to the 2D correlated Wiener process:

Define a vector process driven by the correlated 2D Wiener process:

### Process Properties Extraction(1)

Define a Stratonovich process by its stochastic differential equation:

Available Stratonovich process properties:

Drift and diffusion of the process:

Kolmogorov forward equation:

Inactive is used here to avoid expanding the partial derivatives; use Activate to expand the expression:

Kolmogorov backward equation:

Compute the Stratonovich derivative of a function . The output is a list consisting of drift and diffusion terms:

### Special Stratonovich Processes(5)

A Stratonovich process corresponding to the WienerProcess:

A Stratonovich process corresponding to the GeometricBrownianMotionProcess:

A Stratonovich process corresponding to the BrownianBridgeProcess:

A Stratonovich process corresponding to the OrnsteinUhlenbeckProcess:

A Stratonovich process corresponding to the CoxIngersollRossProcess:

## Applications(3)

Define a vector process corresponding to iterated Stratonovich integrals , , , , , :

Compute its mean function:

And its covariance function:

The dynamics of a free particle under the effect of thermal fluctuation can be modeled by the Langevin equation of motion, , where is the standard WienerProcess and is the strength of the thermal noise. Here it is assumed that can only depend on and focus on the equation of velocity. There are two common ways to integrate the equation of motion: Ito formulation and Stratonovich formulation:

When is a constant, the two formulations are identical and lead to the same stationary distribution as :

If is velocity dependent, then due to the nature of the WienerProcess, has nonzero quadratic variation and the two formulations lead to different results. Convert Ito formulation to the equivalent Stratonovich formulation:

The drift under Stratonovich formulation is different from the drift under Ito formulation:

Create an OrnsteinUhlenbeckProcess and represent it with StratonovichProcess:

Obtain Kolmogorov forward equation:

Solve the equation numerically in with a localized initial condition at and Dirichlet boundary conditions:

Plot the solution of Kolmogorov forward equation at and compare it with the closed-form density function:

Visualize the dynamic of the solution with Animate:

## Properties & Relations(1)

Convert ItoProcess to StratonovichProcess:

Convert back:

## Possible Issues(2)

StratonovichProcess does not support random initial conditions, so cannot be represented:

But it supports processes with fixed initial condition:

Initial time of the driven process needs to match with StratonovichProcess:

With matching initial time, this can be represented: